It is shown that e <mrow class="MJX-TeXAtom-ORD"> 11 12 </

Joshua Foley

Joshua Foley

Answered question

2022-07-05

It is shown that
e 11 12 n n + 1 / 2 e n < n ! < e n n + 1 / 2 e n .
The question is, how to conclude from here that the limit
lim n n ! n n + 1 / 2 e n

Answer & Explanation

Alexia Hart

Alexia Hart

Beginner2022-07-06Added 19 answers

You can prove the monotonicity as follows. Let a n = n ! n n + 1 / 2 e n Then
a n a n + 1 = ( 1 + 1 n ) n + 1 / 2 1 e = exp ( ( n + 1 2 ) log ( 1 + 1 n ) 1 ) .
But
( n + 1 2 ) log ( 1 + 1 n ) 1 = ( 2 n + 1 ) 1 2 log ( 1 + 1 2 n + 1 1 1 2 n + 1 ) 1 = ( 2 n + 1 ) tanh 1 ( 1 2 n + 1 ) 1 ,
and tanh 1 ( x ) = x + x 3 3 + x 5 5 + > x for 0 < x < 1, i.e.,
( n + 1 2 ) log ( 1 + 1 n ) 1 > ( 2 n + 1 ) 1 2 n + 1 1 = 0.
Consequently,
a n a n + 1 > e 0 = 1 ,
i.e., the sequence a n is strictly decreasing.

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